Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Cambridge University Press eBooks

One-dimensional models

K. H. Fischer, John Hertz

Published May 2, 1991• 1 citation
Publisher

Abstract

As in many problems, one-dimensional models can give some insight into the physics of spin glasses because they are easier to solve than those in higher dimensionality, but this advantage can be limited because the physics of the one-dimensional problem may be essentially different. Typically, for example, the higher-dimensional problem we really want to study has a broken symmetry phase at low temperature, while one-dimensional systems (with short-range interactions) do not exhibit broken symmetry at finite temperature. Nevertheless, short-range one-dimensional systems can have broken symmetry at zero temperature, so one can study this relatively simple kind of order and the approach to it as the temperature is lowered to zero. This zero-temperature phase transition can be compared with finite-temperature transitions in higher dimensionality. This is the motivation for Section 12.1, where we will see that the critical point at T = h = 0 in a one-dimensional Ising spin glass with nearest-neighbour interactions has interesting features in common with finite-temperature spin glass transitions, and that they can be calculated analytically, sometimes rather simply. Further insight can be gained by studying one-dimensional systems with long-range (typically power law) interactions. Then if the exponent describing the falloff of the interaction strength with distance is small enough, finite-temperature phase transitions are possible. Though such problems are harder to solve than nearest-neighbour ones, they can still be easier than the full higher-dimensional ones.

DOI
10.1017/cbo9780511628771.013

Related papers

Overcoming System-Size Limitations in Spin Glasses

Helmut G. Katzgraber, Mathias Körner, Frauke Liers +1 more

In order to overcome the limitations of small system sizes in spin-glass simulations, we investigate the one-dimensional Ising spin chain with power-law interactions.The model has the advantage over traditional higher-dimensional Hamiltonians in that a large range of system sizes can be studied.In addition, the universality class of the model can be changed by tuning the power law exponent, thus allowing us to scan from the mean-field to long-range and short-range universality classes.We illustrate the advantages of this model by studying the nature of the spin glass state where our results hint towards a replica symmetry breaking scenario.We also compute ground-state energy distributions and show that mean-field and non-mean-field models are intrinsically different. * )Recently Krzakala and Martin, as well as Palassini and Young, suggest an intermediate picture in which droplets have a fractal surface, and their energy is finite in the thermodynamic limit.See Ref. 5) and references therein for details.* * ) Note that we explicitly distinguish between θ , the stiffness exponent for droplet excitations, and θ, the exponent for domain walls, because DP predicts θ = θ , which we want to test here.

arXiv:cond-mat/0509469•Progress of Theoretical Physics Supplement•Jan 1, 2005•11 citations

The Phase Transitions in a ppp spin Glass Model: A Numerical Study

Prerak Gupta, Auditya Sharma, Bharadwaj Vedula +2 more

We investigate the balanced $M=4$, $p=4$ spin-glass model for a one-dimensional long-range proxy for the finite dimensional short-range $p$-spin glass model to examine the nature of the glass transition beyond mean-field theory. We perform large-scale Monte Carlo equilibrated simulations for both fully connected and power-law diluted versions of the model. The critical temperatures extracted from the finite-size scaling (FSS) analysis of spin-glass susceptibility are in good agreement with theoretical predictions for $σ= 0, 0.25$, and 0.55. For these values of the long-range exponent $σ$ (which is the power of the decrease of the interactions between the spins with their separation), one might have expected that mean-field theory would provide a good description of the system. However, the spin-overlap distribution and the value of the $λ$-parameter do not provide numerical evidence for a one-step replica symmetry breaking (1RSB) phase transition. Instead, our results indicate a direct transition from the paramagnetic state to a full replica symmetry broken phase, with a renormalized value of $λ\equiv ω_2/ω_1 < 1$ suggesting a continuous FRSB transition, despite this ratio being equal to 2 at mean-field level. A value of $λ> 1$ is required for the discontinuous 1RSB transition. We argue that strong finite-size effects and closely spaced transition temperatures remove the expected 1RSB transition for the system sizes which we can study. For values of the exponent $σ= 0.85$, which roughly corresponds to a three dimensional system, we find that the renormalized value of $λ$ is again less than 1, with no signs of either the 1RSB transition or the continuous FRSB transition, suggesting that the Kauzmann temperature $T_K$ in three dimensions might be zero and the complete absence of phase transitions in structural glasses.

arXiv:2604.15433•Apr 16, 2026•0 citations

Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses

Juan Jesus Ruiz-Lorenzo

Spin glasses are the paradigm of complex systems. These materials present really slow dynamics. However, the nature of the spin glass phase in finite dimensional systems is still controversial. Different theories describing the low temperature phase have been proposed: droplet, replica symmetry breaking and chaotic pairs. We present analytical studies of critical properties of spin glasses, in particular, critical exponents at and below the phase transition, existence of a phase transition in a magnetic field, computation of the lower critical dimension (in presence/absence of a magnetic field). We also introduce some rigorous results based on the concept of metastate. Finally, we report some numerical results regarding the construction of the Aizenman-Wehr metastate, scaling of the correlation functions in the spin glass phase and existence of a phase transition in a field, confronting these results with the predictions of different theories.

arXiv:2006.12930•WORLD SCIENTIFIC eBooks•Jun 24, 2020•1 citation

Nature of the spin-glass phase in models with long-range interactions

Matthew Wittmann

Despite decades of effort, our understanding of low-temperature phase of spin glass models with short-range interactions remains incomplete. Replica symmetry breaking (RSB) theory, based on the solution of the Sherrington-Kirkpatrick mean-field model, predicts many pure states; meanwhile, competing theories of short-range systems, such as the droplet picture, predict a single pair of pure states related by time-reversal symmetry, analogously to the ferromagnet. Since RSB certainly holds for the mean-field (infinite-range) model, it is interesting to study short-range models in high dimensions to observe whether RSB also holds here; however, computer simulations of short-range models in high dimensions are difficult because the number of spins to equilibrate grows so rapidly with the linear size of the system.A relatively recent idea which has been fruitful is to instead study one-dimensional models with long-range (power-law) interactions, which are argued to have the same critical behavior as corresponding short-range models in high dimensions, but for which simulations for a range of sizes (crucial for finite-size scaling analysis) are feasible. For these one-dimensional long-range (1DLR) models, we fill in a previously unexplored region of parameter space where the interactions become sufficiently long-range that they must be rescaled with the system size to maintain the thermodynamic limit. We find strong evidence that detailed behavior of the 1DLR models everywhere in this "nonextensive regime" is identical to that of the Sherrington-Kirkpatrick model, lending support to a recent conjecture.In an attempt to distinguish the RSB and droplet pictures, we study recently-proposed observables based on the statistics of individual disorder samples, rather than simply averaging over the disorder as is most frequently done in previous studies. We compare Monte Carlo results for 1DLR models which are proxies for short-range models in 3, 4, and 10 dimensions with previously-obtained data for the 3D and 4D short-range models and the SK model. For one statistic, which is expected to sharply distinguish between the two pictures in the thermodynamic limit, we find that larger system sizes than those currently feasible to simulate are needed to obtain an unambiguous result. We also find that two other recently-proposed statistics, the median of the cumulative overlap distribution and the "typical" overlap distribution, are not particularly helpful in distinguishing between the RSB and droplet pictures.If there are many pure states in the spin-glass phase, we need to carry out some sort of statistical average over them to obtain the thermodynamics. One such prescription for doing this is called the "metastate." Motivated by similarities between the average over pure states specified by the metastate theory and that presumably generated by the nonequilibrium dynamics, we study a 1DLR model which is a proxy for a short-range model in $d=8$ dimensions and measure the evolution of dynamical correlations. We find that the spatial decay of the correlations at distances less than the dynamical correlation length $\\xi(t)$ agrees quantitatively with the predictions of the metastate theory, evaluated according to the RSB picture. We also compute the dynamic exponent defined by $\\xi(t) \\propto t^{1/z(T)}$ and find that it is compatible with the mean-field value of the critical dynamical exponent for short-range spin glasses.Finally, we present a unified view of finite-size scaling (FSS) in dimensions $d$ above the upper critical dimension $d_u$, for both free and periodic boundary conditions. For $d>d_u$, a dangerous irrelevant variable is responsible for both the violation of hyperscaling and the violation of "standard" FSS. We find that the modified hyperscaling proposed to allow for this applies only to $\\vec{k}=\\vec{0}$ fluctuations, while standard FSS applies to $\\vec{k}\\neq\\vec{0}$ fluctuations. Hence the exponent $\\eta$ describing the power-law decay of correlations at criticality is unambiguously $\\eta=0$. With free boundary conditions, the finite-size "shift" is greater than the rounding. Nonetheless, using $T-T_L$, where $T_L$ is the finite-size pseudocritical temperature, as the scaling variable, the data do collapse onto a scaling form that includes the behavior both at $T_L$, where the susceptibility $\\chi$ diverges like $L^{d/2}$, and the bulk $T_c$, where it diverges like $L^2$. We support these claims with data from large-scale simulations of the five-dimensional Ising model.

•Jan 1, 2015•3 citations

Ising Chain with Competing Interactions

John F. Nagle

The one-dimensional spin-\textonehalf{} Ising model is solved exactly with very long-range, equivalent-neighbor ferromagnetic interactions of strength ${J}_{\mathrm{LR}}$ superimposed on nearest-neighbor anti-ferromagnetic interactions of strength $\ensuremath{-}{J}_{\mathrm{sr}}$. For $\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}} >R=3.1532$... there is a critical point of classical type in zero field. For $2\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}=R$ the critical exponents take the values $\ensuremath{\beta}=\frac{1}{4}$, $\ensuremath{\gamma}=1={\ensuremath{\gamma}}^{\ensuremath{'}}$, $\ensuremath{\delta}=5$, and $0=\ensuremath{\alpha}\ensuremath{\ne}{\ensuremath{\alpha}}^{\ensuremath{'}}=\frac{1}{2}$. For $2<\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}<R$ the transition in zero field is of first order, and the critical points which occur in finite field are of classical type. For $0<\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}<2$ there is no phase transition in zero field, although there are classical critical points in a field.

Physical Review A•Nov 1, 1970•107 citations

Beyond mean field theory

K. H. Fischer, John Hertz

Renormalization in pure systems In Section 2.2 we reviewed the basic facts about broken symmetry and phase transitions in pure systems, within mean field theory. This helped set the stage for the mean field description of spin glasses that we studied in Chapters 3–6. We will now want to study the phase transition and the nature of the low-temperature phase for spin glasses with fairly short-ranged interactions in low dimensionality (2 or 3). This situation is far from the region where mean field theory is a good guide, so we will have to learn new methods. Again, we start by learning about them in pure systems. Renormalization is a very general approach to problems with many strongly interacting degrees of freedom. In a statistical mechanical problem, the basic idea is to carry out the trace in the partition function over some of the variables, leaving a new problem with fewer degrees of freedom. One follows how the parameters in the Hamiltonian of the system change as this procedure is iterated many times. Generally, the partial trace corresponds to removing the short-distance degrees of freedom, so each successive effective Hamiltonian has a larger lattice constant (or equivalent microscopic length) than its predecessor: each renormalization step corresponds to a change of scale. Different thermodynamic phases are identified with flows to different fixed points in the parameter space of the Hamiltonian under this semigroup.

Cambridge University Press eBooks•May 2, 1991•0 citations
As in many problems, one-dimensional models can give some insight into the physics of spin glasses because they are easier to solve than those in higher dimensionality, but this advantage can be limited because the physics of the one-dimensional problem may be essentially different. Typically, for example, the higher-dimensional problem we really want to study has a broken symmetry phase at low temperature, while one-dimensional systems (with short-range interactions) do not exhibit broken symmetry at finite temperature. Nevertheless, short-range one-dimensional systems can have broken symmetry at zero temperature, so one can study this relatively simple kind of order and the approach to it as the temperature is lowered to zero. This zero-temperature phase transition can be compared with finite-temperature transitions in higher dimensionality. This is the motivation for Section 12.1, where we will see that the critical point at T = h = 0 in a one-dimensional Ising spin glass with nearest-neighbour interactions has interesting features in common with finite-temperature spin glass transitions, and that they can be calculated analytically, sometimes rather simply. Further insight can be gained by studying one-dimensional systems with long-range (typically power law) interactions. Then if the exponent describing the falloff of the interaction strength with distance is small enough, finite-temperature phase transitions are possible. Though such problems are harder to solve than nearest-neighbour ones, they can still be easier than the full higher-dimensional ones.
In order to overcome the limitations of small system sizes in spin-glass simulations, we investigate the one-dimensional Ising spin chain with power-law interactions.The model has the advantage over traditional higher-dimensional Hamiltonians in that a large range of system sizes can be studied.In addition, the universality class of the model can be changed by tuning the power law exponent, thus allowing us to scan from the mean-field to long-range and short-range universality classes.We illustrate the advantages of this model by studying the nature of the spin glass state where our results hint towards a replica symmetry breaking scenario.We also compute ground-state energy distributions and show that mean-field and non-mean-field models are intrinsically different. * )Recently Krzakala and Martin, as well as Palassini and Young, suggest an intermediate picture in which droplets have a fractal surface, and their energy is finite in the thermodynamic limit.See Ref. 5) and references therein for details.* * ) Note that we explicitly distinguish between θ , the stiffness exponent for droplet excitations, and θ, the exponent for domain walls, because DP predicts θ = θ , which we want to test here.
We investigate the balanced M=4M=4M=4, p=4p=4p=4 spin-glass model for a one-dimensional long-range proxy for the finite dimensional short-range ppp-spin glass model to examine the nature of the glass transition beyond mean-field theory. We perform large-scale Monte Carlo equilibrated simulations for both fully connected and power-law diluted versions of the model. The critical temperatures extracted from the finite-size scaling (FSS) analysis of spin-glass susceptibility are in good agreement with theoretical predictions for σ=0,0.25σ= 0, 0.25σ=0,0.25, and 0.55. For these values of the long-range exponent σσσ (which is the power of the decrease of the interactions between the spins with their separation), one might have expected that mean-field theory would provide a good description of the system. However, the spin-overlap distribution and the value of the λλλ-parameter do not provide numerical evidence for a one-step replica symmetry breaking (1RSB) phase transition. Instead, our results indicate a direct transition from the paramagnetic state to a full replica symmetry broken phase, with a renormalized value of λ≡ω2/ω1<1λ\equiv ω_2/ω_1 < 1λ≡ω2​/ω1​<1 suggesting a continuous FRSB transition, despite this ratio being equal to 2 at mean-field level. A value of λ>1λ> 1λ>1 is required for the discontinuous 1RSB transition. We argue that strong finite-size effects and closely spaced transition temperatures remove the expected 1RSB transition for the system sizes which we can study. For values of the exponent σ=0.85σ= 0.85σ=0.85, which roughly corresponds to a three dimensional system, we find that the renormalized value of λλλ is again less than 1, with no signs of either the 1RSB transition or the continuous FRSB transition, suggesting that the Kauzmann temperature TKT_KTK​ in three dimensions might be zero and the complete absence of phase transitions in structural glasses.
Spin glasses are the paradigm of complex systems. These materials present really slow dynamics. However, the nature of the spin glass phase in finite dimensional systems is still controversial. Different theories describing the low temperature phase have been proposed: droplet, replica symmetry breaking and chaotic pairs. We present analytical studies of critical properties of spin glasses, in particular, critical exponents at and below the phase transition, existence of a phase transition in a magnetic field, computation of the lower critical dimension (in presence/absence of a magnetic field). We also introduce some rigorous results based on the concept of metastate. Finally, we report some numerical results regarding the construction of the Aizenman-Wehr metastate, scaling of the correlation functions in the spin glass phase and existence of a phase transition in a field, confronting these results with the predictions of different theories.
Despite decades of effort, our understanding of low-temperature phase of spin glass models with short-range interactions remains incomplete. Replica symmetry breaking (RSB) theory, based on the solution of the Sherrington-Kirkpatrick mean-field model, predicts many pure states; meanwhile, competing theories of short-range systems, such as the droplet picture, predict a single pair of pure states related by time-reversal symmetry, analogously to the ferromagnet. Since RSB certainly holds for the mean-field (infinite-range) model, it is interesting to study short-range models in high dimensions to observe whether RSB also holds here; however, computer simulations of short-range models in high dimensions are difficult because the number of spins to equilibrate grows so rapidly with the linear size of the system.A relatively recent idea which has been fruitful is to instead study one-dimensional models with long-range (power-law) interactions, which are argued to have the same critical behavior as corresponding short-range models in high dimensions, but for which simulations for a range of sizes (crucial for finite-size scaling analysis) are feasible. For these one-dimensional long-range (1DLR) models, we fill in a previously unexplored region of parameter space where the interactions become sufficiently long-range that they must be rescaled with the system size to maintain the thermodynamic limit. We find strong evidence that detailed behavior of the 1DLR models everywhere in this "nonextensive regime" is identical to that of the Sherrington-Kirkpatrick model, lending support to a recent conjecture.In an attempt to distinguish the RSB and droplet pictures, we study recently-proposed observables based on the statistics of individual disorder samples, rather than simply averaging over the disorder as is most frequently done in previous studies. We compare Monte Carlo results for 1DLR models which are proxies for short-range models in 3, 4, and 10 dimensions with previously-obtained data for the 3D and 4D short-range models and the SK model. For one statistic, which is expected to sharply distinguish between the two pictures in the thermodynamic limit, we find that larger system sizes than those currently feasible to simulate are needed to obtain an unambiguous result. We also find that two other recently-proposed statistics, the median of the cumulative overlap distribution and the "typical" overlap distribution, are not particularly helpful in distinguishing between the RSB and droplet pictures.If there are many pure states in the spin-glass phase, we need to carry out some sort of statistical average over them to obtain the thermodynamics. One such prescription for doing this is called the "metastate." Motivated by similarities between the average over pure states specified by the metastate theory and that presumably generated by the nonequilibrium dynamics, we study a 1DLR model which is a proxy for a short-range model in d=8d=8d=8 dimensions and measure the evolution of dynamical correlations. We find that the spatial decay of the correlations at distances less than the dynamical correlation length xi(t)\\xi(t)xi(t) agrees quantitatively with the predictions of the metastate theory, evaluated according to the RSB picture. We also compute the dynamic exponent defined by xi(t)proptot1/z(T)\\xi(t) \\propto t^{1/z(T)}xi(t)proptot1/z(T) and find that it is compatible with the mean-field value of the critical dynamical exponent for short-range spin glasses.Finally, we present a unified view of finite-size scaling (FSS) in dimensions ddd above the upper critical dimension dud_udu​, for both free and periodic boundary conditions. For d>dud>d_ud>du​, a dangerous irrelevant variable is responsible for both the violation of hyperscaling and the violation of "standard" FSS. We find that the modified hyperscaling proposed to allow for this applies only to veck=vec0\\vec{k}=\\vec{0}veck=vec0 fluctuations, while standard FSS applies to veckneqvec0\\vec{k}\\neq\\vec{0}veckneqvec0 fluctuations. Hence the exponent eta\\etaeta describing the power-law decay of correlations at criticality is unambiguously eta=0\\eta=0eta=0. With free boundary conditions, the finite-size "shift" is greater than the rounding. Nonetheless, using T−TLT-T_LT−TL​, where TLT_LTL​ is the finite-size pseudocritical temperature, as the scaling variable, the data do collapse onto a scaling form that includes the behavior both at TLT_LTL​, where the susceptibility chi\\chichi diverges like Ld/2L^{d/2}Ld/2, and the bulk TcT_cTc​, where it diverges like L2L^2L2. We support these claims with data from large-scale simulations of the five-dimensional Ising model.
The one-dimensional spin-\textonehalf{} Ising model is solved exactly with very long-range, equivalent-neighbor ferromagnetic interactions of strength JLR{J}_{\mathrm{LR}}JLR​ superimposed on nearest-neighbor anti-ferromagnetic interactions of strength $\ensuremath{-}{J}_{\mathrm{sr}}$. For $\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}} >R=3.1532$... there is a critical point of classical type in zero field. For $2\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}=R$ the critical exponents take the values $\ensuremath{\beta}=\frac{1}{4}$, $\ensuremath{\gamma}=1={\ensuremath{\gamma}}^{\ensuremath{'}}$, $\ensuremath{\delta}=5$, and $0=\ensuremath{\alpha}\ensuremath{\ne}{\ensuremath{\alpha}}^{\ensuremath{'}}=\frac{1}{2}$. For $2<\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}<R$ the transition in zero field is of first order, and the critical points which occur in finite field are of classical type. For $0<\ensuremath{-}\frac{{J}_{\mathrm{lr}}}{{J}_{\mathrm{sr}}}<2$ there is no phase transition in zero field, although there are classical critical points in a field.
Renormalization in pure systems In Section 2.2 we reviewed the basic facts about broken symmetry and phase transitions in pure systems, within mean field theory. This helped set the stage for the mean field description of spin glasses that we studied in Chapters 3–6. We will now want to study the phase transition and the nature of the low-temperature phase for spin glasses with fairly short-ranged interactions in low dimensionality (2 or 3). This situation is far from the region where mean field theory is a good guide, so we will have to learn new methods. Again, we start by learning about them in pure systems. Renormalization is a very general approach to problems with many strongly interacting degrees of freedom. In a statistical mechanical problem, the basic idea is to carry out the trace in the partition function over some of the variables, leaving a new problem with fewer degrees of freedom. One follows how the parameters in the Hamiltonian of the system change as this procedure is iterated many times. Generally, the partial trace corresponds to removing the short-distance degrees of freedom, so each successive effective Hamiltonian has a larger lattice constant (or equivalent microscopic length) than its predecessor: each renormalization step corresponds to a change of scale. Different thermodynamic phases are identified with flows to different fixed points in the parameter space of the Hamiltonian under this semigroup.